In the final episode of the 3rd season when Oliver says "I think I already told everyone to call you Speedy", I thought he was being a wiseass. Yet in the 4th season I've frequently heard him call her that when they've "suited up". Is this really the handle Thea's been going by?
Yes, her codename is really Speedy. In-universe, it's Oliver's old nickname for her, from when they were kids:
Oliver: No one's called me [Ollie] in a while, Speedy.
Thea: Worst nickname ever.
Oliver: What, always chasing after me as a kid? I thought it fit pretty well.
Arrow Season 1 Episode 1: "Pilot"
Out of universe, it's a reference to the legacy Green Arrow sidekick. Speedy was one of Roy Harper's many aliases; along with "Arsenal" (the one they ended up using on Arrow) and the incredibly imaginative "Red Arrow".